1.3 Powers and Standard Form
Understanding Powers (Exponents)
Powers (or exponents) are a way of expressing repeated multiplication. For example, instead of writing 2 × 2 × 2 × 2, we can write 2⁴ (2 to the power of 4). The small number above and to the right of the base number is called the index or exponent, and it tells us how many times to multiply the base number by itself.
Roots are the opposite of powers. The square root (√) of a number is a number that, when multiplied by itself, gives the original number. The cube root (∛) of a number is a number that, when multiplied by itself three times, gives the original number.
Indices are just the number that tells you the power. Fractional indices represent roots:
Powers (or exponents) are a way of expressing repeated multiplication. For example, instead of writing 2 × 2 × 2 × 2, we can write 2⁴ (2 to the power of 4). The small number above and to the right of the base number is called the index or exponent, and it tells us how many times to multiply the base number by itself.
Basic Examples:
• 3² = 3 × 3 = 9
• 5³ = 5 × 5 × 5 = 125
• 10⁴ = 10 × 10 × 10 × 10 = 10,000
Rules for Working with Powers• 3² = 3 × 3 = 9
• 5³ = 5 × 5 × 5 = 125
• 10⁴ = 10 × 10 × 10 × 10 = 10,000
Rule 1: Multiplying Powers (Same Base)
Add the indices: am × an = a(m+n)
Add the indices: am × an = a(m+n)
Example: 2³ × 2⁴ = 2(3+4) = 2⁷ = 128
Rule 2: Dividing Powers (Same Base)
Subtract the indices: am ÷ an = a(m-n)
Subtract the indices: am ÷ an = a(m-n)
Example: 5⁶ ÷ 5² = 5(6-2) = 5⁴ = 625
Rule 3: Power of a Power
Multiply the indices: (am)n = a(m×n)
Multiply the indices: (am)n = a(m×n)
Example: (3²)³ = 3(2×3) = 3⁶ = 729
Rule 4: Any Number to the Power of 0
a⁰ = 1 (for any non-zero value of a)
a⁰ = 1 (for any non-zero value of a)
Example: 7⁰ = 1, 100⁰ = 1
Rule 5: Negative Indices
Negative indices represent reciprocal powers: a-n = 1/(an)
Understanding RootsNegative indices represent reciprocal powers: a-n = 1/(an)
Example: 2-3 = 1/(2³) = 1/8 = 0.125
Roots are the opposite of powers. The square root (√) of a number is a number that, when multiplied by itself, gives the original number. The cube root (∛) of a number is a number that, when multiplied by itself three times, gives the original number.
Root Examples:
• √16 = 4 because 4 × 4 = 16
• ∛8 = 2 because 2 × 2 × 2 = 8
• √25 = 5 because 5 × 5 = 25
• ∛27 = 3 because 3 × 3 × 3 = 27
Fractional Indices• √16 = 4 because 4 × 4 = 16
• ∛8 = 2 because 2 × 2 × 2 = 8
• √25 = 5 because 5 × 5 = 25
• ∛27 = 3 because 3 × 3 × 3 = 27
Indices are just the number that tells you the power. Fractional indices represent roots:
Fractional Index Rules:
• x1/2 = √x (square root)
• x1/3 = ∛x (cube root)
• x3/2 = (√x)³ or √(x³)
• x2/3 = (∛x)² or ∛(x²)
• x1/2 = √x (square root)
• x1/3 = ∛x (cube root)
• x3/2 = (√x)³ or √(x³)
• x2/3 = (∛x)² or ∛(x²)
Example 1: 161/2 = √16 = 4
Example 2: 81/3 = ∛8 = 2
Example 3: 272/3 = (∛27)² = 3² = 9
Example 2: 81/3 = ∛8 = 2
Example 3: 272/3 = (∛27)² = 3² = 9
🧮 Power Calculator
Base:
Exponent:
Base:
Exponent:
🎯 Match the Rule: Which rule applies to the equation below?
Real Life Uses:
• Computing: Data storage capacities (kilobytes, megabytes, gigabytes use powers of 2)
• Science: Exponential growth/decay in populations, radioactive materials, and bacteria
• Finance: Compound interest calculations use powers to calculate growth over time
• Physics: Inverse square laws (gravity, light intensity) use negative powers
• Engineering: Calculating volumes and surface areas of complex shapes
• Music: Frequency ratios between musical notes involve fractional powers
• Computing: Data storage capacities (kilobytes, megabytes, gigabytes use powers of 2)
• Science: Exponential growth/decay in populations, radioactive materials, and bacteria
• Finance: Compound interest calculations use powers to calculate growth over time
• Physics: Inverse square laws (gravity, light intensity) use negative powers
• Engineering: Calculating volumes and surface areas of complex shapes
• Music: Frequency ratios between musical notes involve fractional powers
What is Standard Form?
Standard form makes writing very large numbers, or numbers with lots of decimal places easy, by writing them as a number between 1 and 10 multiplied by a power of 10.
It is much easier to say that the Earth is 2.5 × 10²⁰m away from the center of our galaxy instead of 250,000,000,000,000,000,000m, or that an atom is 3.0 × 10⁻¹⁰m across instead of 0.0000000003m.
How to Write Numbers in Standard Form
The format is: a × 10n where:
• a is a number between 1 and 10 (but not 10 itself)
• n is positive for numbers greater than 10
• n is negative for numbers less than 1
Standard form makes writing very large numbers, or numbers with lots of decimal places easy, by writing them as a number between 1 and 10 multiplied by a power of 10.
It is much easier to say that the Earth is 2.5 × 10²⁰m away from the center of our galaxy instead of 250,000,000,000,000,000,000m, or that an atom is 3.0 × 10⁻¹⁰m across instead of 0.0000000003m.
How to Write Numbers in Standard Form
The format is: a × 10n where:
• a is a number between 1 and 10 (but not 10 itself)
• n is positive for numbers greater than 10
• n is negative for numbers less than 1
Converting Large Numbers (n is positive):
Example 1: Convert 4,500 to standard form
Step 1: Move decimal left until you have a number between 1 and 10 → 4.5
Step 2: Count places moved → 3 places
Step 3: Write as 4.5 × 10³ = 4.5 × 10³
Example 2: Convert 123,000,000 to standard form
Step 1: Move decimal → 1.23
Step 2: Count places moved → 8 places
Step 3: Write as 1.23 × 10⁸
Step 1: Move decimal left until you have a number between 1 and 10 → 4.5
Step 2: Count places moved → 3 places
Step 3: Write as 4.5 × 10³ = 4.5 × 10³
Example 2: Convert 123,000,000 to standard form
Step 1: Move decimal → 1.23
Step 2: Count places moved → 8 places
Step 3: Write as 1.23 × 10⁸
Converting Small Numbers (n is negative):
Calculating with Standard Form
Example 1: Convert 0.0067 to standard form
Step 1: Move decimal right until you have a number between 1 and 10 → 6.7
Step 2: Count places moved → 3 places
Step 3: Write as 6.7 × 10⁻³
Example 2: Convert 0.000045 to standard form
Step 1: Move decimal → 4.5
Step 2: Count places moved → 5 places
Step 3: Write as 4.5 × 10⁻⁵
Step 1: Move decimal right until you have a number between 1 and 10 → 6.7
Step 2: Count places moved → 3 places
Step 3: Write as 6.7 × 10⁻³
Example 2: Convert 0.000045 to standard form
Step 1: Move decimal → 4.5
Step 2: Count places moved → 5 places
Step 3: Write as 4.5 × 10⁻⁵
Multiplication in Standard Form
Multiply the front numbers, add the powers
Multiply the front numbers, add the powers
Example: (2 × 10³) × (3 × 10⁴)
Step 1: Multiply front numbers → 2 × 3 = 6
Step 2: Add powers → 3 + 4 = 7
Step 3: Result = 6 × 10⁷
Step 1: Multiply front numbers → 2 × 3 = 6
Step 2: Add powers → 3 + 4 = 7
Step 3: Result = 6 × 10⁷
Division in Standard Form
Divide the front numbers, subtract the powers
Divide the front numbers, subtract the powers
Example: (6 × 10⁵) ÷ (2 × 10²)
Step 1: Divide front numbers → 6 ÷ 2 = 3
Step 2: Subtract powers → 5 - 2 = 3
Step 3: Result = 3 × 10³
Step 1: Divide front numbers → 6 ÷ 2 = 3
Step 2: Subtract powers → 5 - 2 = 3
Step 3: Result = 3 × 10³
Powers of Standard Form
Apply the power to both parts
Apply the power to both parts
Example: (4 × 10⁶)²
Step 1: Square the front number → 4² = 16
Step 2: Multiply the power → 6 × 2 = 12
Step 3: Result = 16 × 10¹²
Step 4: Adjust to proper form = 1.6 × 10¹³
Step 1: Square the front number → 4² = 16
Step 2: Multiply the power → 6 × 2 = 12
Step 3: Result = 16 × 10¹²
Step 4: Adjust to proper form = 1.6 × 10¹³
Addition & Subtraction in Standard Form:
Important: Powers of 10 must be the same before adding/subtracting
Same Powers Example:
• (2 × 10³) + (3 × 10³) = 5 × 10³
• (5 × 10⁴) - (2 × 10⁴) = 3 × 10⁴
Different Powers Example:
(4 × 10⁵) + (2 × 10⁶)
Step 1: Convert to same power → (0.4 × 10⁶) + (2 × 10⁶)
Step 2: Add front numbers → 0.4 + 2 = 2.4
Step 3: Result = 2.4 × 10⁶
Same Powers Example:
• (2 × 10³) + (3 × 10³) = 5 × 10³
• (5 × 10⁴) - (2 × 10⁴) = 3 × 10⁴
Different Powers Example:
(4 × 10⁵) + (2 × 10⁶)
Step 1: Convert to same power → (0.4 × 10⁶) + (2 × 10⁶)
Step 2: Add front numbers → 0.4 + 2 = 2.4
Step 3: Result = 2.4 × 10⁶
🧮 Standard Form Converter
Enter a number:
Enter a number:
🧮 Standard Form Calculator: Multiply or Divide
( × 10^)
( × 10^)
🎯 Size Comparison: Drag to order from smallest to largest
Real Life Uses:
• Astronomy: Distances between planets, stars, and galaxies (light years = 9.46 × 10¹⁵m)
• Physics: Speed of light (3 × 10⁸ m/s), Planck's constant (6.63 × 10⁻³⁴ J·s)
• Chemistry: Avogadro's number (6.02 × 10²³ molecules/mol), atomic masses
• Biology: Size of cells, bacteria, and viruses (typically 10⁻⁶ to 10⁻⁹ meters)
• Computing: Number of transistors in processors, data processing speeds
• Economics: National debts, GDP, and global financial figures
• Medicine: Drug concentrations and dosages in bloodstream
• Astronomy: Distances between planets, stars, and galaxies (light years = 9.46 × 10¹⁵m)
• Physics: Speed of light (3 × 10⁸ m/s), Planck's constant (6.63 × 10⁻³⁴ J·s)
• Chemistry: Avogadro's number (6.02 × 10²³ molecules/mol), atomic masses
• Biology: Size of cells, bacteria, and viruses (typically 10⁻⁶ to 10⁻⁹ meters)
• Computing: Number of transistors in processors, data processing speeds
• Economics: National debts, GDP, and global financial figures
• Medicine: Drug concentrations and dosages in bloodstream